Fast computations of the exponential function

Abstract

In this paper we present an algorithm which shows that the exponential function has algebraic complexity O(log2 n), i.e., can be evaluated with relative error O(2-n) using O(log2 n) infinite-precision additions, subtractions, multiplications and divisions. This solves a question of J. M. Borwein and P. B. Borwein [9]. The best known lower bound for the algebraic complexity of the exponential function is Ω(log n). The best known upper and lower bounds for the bit complexity of the exponential function are O(μ(n) log n) [10] and Ω (ν (n)) [4], respectively, where μ (n) denotes an upper bound and ν (n) denotes a lower bound for the bit complexity of n-bit integer multiplication. The presented algorithm has bit complexity O(μ (n) log n).